What will happen with Blender 2.8 game engine?
As far as we know the Blender Game engine has been removed from official Blender releases for good, at least in its old form. As of commit 159806140fd3 - Removing Blender Game Engine from Blender 2.8 on April 2018, BGE is no more, along with other &quot;aging&quot; components like the Blender Internal Render Engine viewport drawing code among others.
Ton Roosendaal himself was given the honor of making the commit that put the old code to rest.

As for the reasons for removal we can conclude it was an old and aging component, it was largely unmaintained, suffered from several bugs and limitations, and the Blender Foundation probably lacked the energy, motivation and manpower to maintain it properly.

Adding new things to Blender gets much easier that way. The Blender
Internal code started in early 90s. Over 25 years old! Functionally,
EEVEE can (and will) completely replace it.
Ton Roosendaal

It was also very architecturally outdated, lagged severely behind the current industry standards and modern technologies in many ways, it would be very hard to bring it up to date. Modernizing it or introducing new features would end up being more work than a full rewrite from scratch.
For that reason they decided to release Blender from these &quot;metaphorical shackles&quot; holding it back; making it easier to move forward and update other more important and frequently used features.
This frees Blender developers to work on other things, allowing more manpower to be redirected to truly useful new features like a better viewport drawing code, a modern layer system, a better dependency graph, usability improvements, among others. These will instead help you produce better game assets (which is Blender's main focus, and where its greatest strength lies anyway), rather than trying to run the game itself, leaving that role to dedicated engines.
&quot;Why wont they make a new one then?&quot;
Probably both because &quot;we are too late to the race&quot; and it would be a tremendous amount of work to write one well. There are already a number of well established third party game engine solutions out there, both free, open source, or comercial; and competing with those would be extremely hard and pointless.
Writing a full game engine is no small task, it is a full time job on its own, and maintaining it well would require a lot of continued manpower and resources. Since expectations are very high (supporting multitude of hardware and environments, multi-platform publishing, modern physics, logic systems, networking, security, smooth performance, and all the modern commodities expected of contemporary game engines) it would probably always end up feeling inferior or lacking in lots of ways compared to more mature established competitors.
What About Forking It?
UPBGE is a third party fork, whose objective is updating and modernizing the BGE with better technologies and features, with the end goal of eventually merging it back with main Blender development branch.
I fear its future is uncertain at this point unfortunately. While I believe Blender developers won't actively do anything to purposefully break Blender trunk codebase compatibility, I speculate it will naturally become increasingly hard for UPBGE developers to maintain their code working, as features are removed and main Blender development deviates further from the current design.
UPBGE will probably continue to work as standalone based against current 2.79 versions for any foreseeable future, but you will likely miss all the other improvements made in subsequent versions.
While the final decision is always yours, as it stands your time is probably best spent learning something else more ubiquitous and modern, giving you a more lasting and future proof investment.
So, is it really gone for good?
In its current form yes.
There have been talks about replacing the old game engine with a newer better integrated &quot;Interactive Mode&quot;. It would be a more integral part of Blender, as opposed to a separate component with a lot of duplicated code that re-implemented a very limited subset of supported features we currently have.
This is not meant to be a direct replacement of the old BGE, nor a discrete game engine in the traditional sense with wide publishing capabilities, rather more of an integrated real-time &quot;presentation tool&quot; or interaction mode with a physics simulation sandbox environment, that runs directly inside Blender's viewport on currently supported platforms.
On 28 May 2018 Ton Roosendaal announced in the Developers Mailing list that a portion of the 2.8 Code Quest funds were reserved for the development of a new and improved real time rendering system or &quot;Interactive Mode&quot; and Benoit Bolsee, a developer historically known for his involvement in real time side of Blender, accepted a grant to work part time on this for one year.
I speculate this will probably feature a new node based workflow and logic system,  real time physics, and integrate well with other upcoming features like &quot;Object Nodes&quot; also known as &quot;Everything Nodes&quot; in one way or another.
When will it be ready?
A project of this magnitude requires a lot of careful planing and design to get done right, this will lay solid foundations on which to work on. On top of that there is still the huge amount of work required to actually implement it and get it working, tested and stable enough for the general public.
This would take a dedicated team working full time a long time to get done, it will take even more for a single person to do it alone.
Since Benoit is only working on it part time and for a relatively short period of one year, this is likely still very far away in the horizon.
As of 2020 development seems to be indefinitely on standby, probably waiting on  solid planning, development of required dependencies and maturing of current features. There are currently no predictions of when or if it will ever be resumed.
If you would like to help and can't code, you can contribute to the Development Fund. This will allow hiring more full time developers to work on newer features. Have in mind that there is no guarantee this will specifically direct funds dedicated to Interaction Mode, but they might get there faster. And if there is enough interest maybe a full time developer slots can be allocated for these tasks.
So what do I do now?
Though BGE is going away for future versions nothing stops you from continuing using it under Blender 2.79 or UPBGE for the foreseeable future, as long as they work in current hardware and operating systems.
If you are spending time and resources learning the craft, you might as well invest it on something popular and well accepted in the industry, with a bigger audience and wider reach. Learning resources will likely be more abundant, and it might even increase your future employability, if you seek to work in the industry.
There are many third party game engines out there, good for many different purposes and workflows, and for varying &quot;wallet sizes&quot;. Some integrate very well with Blender, even using its native UI (like the Node Editor through PyNodes) for editing a varying subset of supported features and functionality.
They also provide extended publishing capabilities, reaching many platforms never before supported by Blender Game Engine (like mobile devices, web, cross platform development, among others).
The best known and most well established ones are probably:

Unreal Engine
Unity
Uniengine - Focus on real time presentations for industry and enterprises
CryEngine
Lumberyard - Amazon's Open Source version of the CryEngine

There are also free and open source solutions that integrate very well into Blender pipeline and native UI like:

Godot - Well established free alternative, 2D graphics focus, popular for mobile content.
Armory 3D - Relatively young entry, actively maintained, free and open source license.
Urho3D - As pointed out by user Kabu in the comments
Ogre 3D -  One of the oldest open source engines around, since 2001

While not a game engines per-se, products like Blend4Web and Verge3D provide Blender addons for web publishing that also have some form of node-based logic systems and scripting capabilities. These allow creation of basic behaviors, and a certain degrees of interactivity, thus providing a simplistic yet relatively capable web browser based game creation solution.